Math 8 Homework 9 Solutions
1
Arithmetic and Algebra of Complex Numbers
√
(a) (a) (1 + i)100 = ( 2eiπ/4 )100 = 250 .
(b) (1 + eiθ )n = einθ/2 (2n cosn θ/2).
(c) exp(eiθ ) = exp(cos θ + i sin θ) = ecos θ ei sin θ = ecos θ (cos(sin θ) + i sin(sin θ))
(b)
(i) The set of all points 4 units away from 1 is a circle of radius 4 centered at 1.
(ii) The set of points equidistant from i and 4 is the perpendicular bisector of the line segment joining i
and 4.
(iii) Given z ∈ C, note that |z| is distance to 0 while Re(z + 2) is the distance to the vertical line y = −2.
The set for which these are equal is a parabola.
√
(c) (i) z 6 − 2z 3 + 2 = 0 gives (z 3 − 1)2 = −1, so that z 3 = 1 ± i. Converting to polar, z 3 = 2e±iπ/4 and
hence z = 21/6 exp(2ikπ/3 ± iπ/12), where k ∈ {0, 1, 2}.
(ii) (z+1)5 = z 5 becomes (1+1/z)5 = 1 upon division. Hence z = 1/(exp(2ikπ/5)−1) with k ∈ {0, 1, 2, 3, 4}.
We exclude k = 0 since it leads to division by 0.
√
(iii) ez = √
1 + i becomes ez = 2eiπ/4 when we rewrite in polar. The real part of z = x + iy must satisfy
ex = √2, so that x = ln(2)/2. The imaginary part must satisfy y = π/4 + 2πk for some k ∈ Z. Thus
z = ln 2 + iπ/4 + 2kiπ with k ∈ Z.
(iv) z 4 = 5(z − 1)(z 2 − z + 1) becomes z 4 − 5z 3 + 10z 2 − 10z + 5 = 0 after rearranging. This is suspiciously
familiar, so multiplying by z and subtracting 1 gives
(z − 1)5 = −1.
That is, z = 1 + e(2k+1)iπ/5 with k ∈ {0, 1, 2, 3, 4}. However, k = 0 leads to z = 0 which doesn’t satisfy
the original equation. So there are only 4 solutions (as can be expected from a quartic equation).
(d) Proof. Rewrite cos(nθ) = Re(einθ ) and evaluate the geometric series:
∞
∞ iθ n
X
X
cos(nθ)
e
1
=
Re
.
= Re
n
2
2
1
−
eiθ /2
n=0
n=0
The rest is algebra.
∞
X
1
2
2(2 − e−iθ )
4 − 2e−iθ
cos(nθ)
=
Re
=
Re
=
Re
=
Re
.
2n
1 − eiθ /2
2 − eiθ
(2 − eiθ )(2 − e−iθ )
5 − 2(eiθ + e−iθ )
n=0
The denominator contains an expression for 2 cos θ:
∞
X
4 − 2 cos θ + 2i sin θ
cos(nθ)
4 − 2 cos θ
=
,
= Re
n
2
5
−
4
cos
θ
5 − 4 cos θ
n=0
and we’re done.
(e) Proof. TYPO ALERT: I should’ve written nonconstant in the problem statement. Let a ∈ C be given. Then
p(z) − a is a nonconstant polynomial, and by the fundamental theorem of algebra, there exists z ∈ C so that
p(z) − a = 0. Hence p(z) = a and p is surjective.
(f) Proof. Define the function
z−p
.
1 − pz
Clearly f (p) = 0. Furthermore, the denominator of f never vanishes; for z ∈ D we have |pz| < 1. Finally,
note that as |z| → 1 we have z → 1/z so that
z−a z−a = |z| = 1,
→
|f (z)| = 1 − za 1 − a/z f (z) =
as desired.
2
Problem Solving with Complex Numbers
(a) Proof. Let ω = exp(2iπ/n). The product of the lengths we want is
P =
n−1
Y
|1 − ω k |.
k=1
Let f (z) = 1 + z + z 2 + · · · + z n−1 , whose roots are ω, ω 2 , . . . , ω n−1 . Then g(z) = f (1 − z) is a polynomial
with roots 1 − ω, 1 − ω 2 , . . . , 1 − ω n−1 . The product of the roots of a polynomial g with degree n − 1 is
(−1)n−1 g(0), so we have
n−1
Y
P =
(1 − ω k ) = |(−1)n−1 g(0)| = |f (1)| = n.
k=1
Now we want to compute
S=
n−1
X
|1 − ω k |2 =
k=1
n−1
X
n−1
X
k=1
k=1
(1 − ω k )(1 − ω n−k ) =
(1 − ω k − ω n−k + ω n ).
The sum ω + ω 2 + · · · + ω n−1 = −1. Using this and the fact that ω n = 1,
S = 2(n − 1) − (−1) − (−1) = 2n,
and we’re done.
(b) Proof. Let a, b, m, n be positive integers. Then we have
(a2 + b2 )(m2 + n2 ) = |a + bi|2 |m + ni|2 = |(am − bn) + i(an + bm)|2 = (am − bn)2 + (an + bm)2 .
Hence the product of two sums of 2 squares is itself a sum of 2 squares.
(c) Proof. Set z 7→ ωz to get
f (ωz) + f (ω 2 z) = exp(ωz).
(1)
In this equation again set z 7→ ωz to find
f (ω 2 z) + f (ω 3 z) = exp(ω 2 z).
The second term above simplifies since ω 3 = 1, so we find
f (ω 2 z) + f (z) = exp(ω 2 z).
(2)
If we take the given identity
f (z) + f (ωz) = exp(z),
subtracting (1), adding (2), and dividing by 3 gives
f (z) =
exp(z) − exp(ωz) + exp(ω 2 z)
.
3
Proving uniqueness of f is immediate. Starting from the identity from f we derived exactly what f must be,
so there can be no other f .
(d) Proof. We begin with a geometric series. When z ∈ D the series
z a + z a+d + z a+2d + · · · = z a (1 + z d + (z d )2 + · · · ) =
is a convergent geometric series.
Now suppose that we find integers ak , dk so that each dk is distinct and
z
z a1
z a2
z an
=
+
+
·
·
·
+
1−z
1 − z d1
1 − z d2
1 − z dn
za
1 − zd
for all z ∈ D. Without loss of generality let d1 be the largest such dk . Multiplying gives
a2
z
z(1 − z d1 )
z an
+
·
·
·
+
= z a1 + (1 − z d1 )
1−z
1 − z d2
1 − z dn
(3)
Now we have to keep z ∈ D, but we can take the limit z → e2iπ/d1 while z stays within the disk. Assuming
d1 6= 1 we deduce
0 = e2ia1 π/d1 ,
which is impossible. Hence the largest dk is 1 and as all dk were distinct, there are no others. That is, n = 1.
Returning to equation (3) we see that z = z a1 , so a1 = 1. This proves the second result.
Finally, assume we can decompose N into a disjoint collection of arithemtic progressions:
N = {a1 , a1 + d1 , a1 + 2d1 , . . .} ∪ · · · ∪ {an , an + dn , an + 2dn , . . .},
with each dk distinct. We can use this partition to write
z + z 2 + z 3 + · · · = (z a1 + z a1 +d1 + · · · ) + · · · + (z an + z an +dn + · · · )
for all z ∈ D. Summing these series gives
z a1
z a2
z an
z
=
+
+
·
·
·
+
.
d
d
1−z
1−z 1
1−z 2
1 − z dn
By the above reasoning, n = a1 = d1 = 1. That is, our partition of N must have been the trivial one.